Global Properties Of Submanifolds In Product Manifolds

In the thesis, we shall study several global properties of submanifolds in some product manifolds and give their applications. The thesis consists of two parts. In the first part(Chapter 3), we explore the Weierstrass representation for surfaces of pre-scribed mean curvature in H2 x R and obtainTheorem 1 ([37]) Let x = (x1, x2, x3):∑→H2×R be an isometric immersion and G1:∑→U1 be the normal Gauss map, where U1 is defined in (3.5). Then we have, on U1,Theorem 2 ([37]) Let E be a simply connected Riemann surface, H:∑→R be a C1-function, and G1:∑→U1 be a smooth mapping, where U1 is defined in (3.5). Assume that G1 satisfies the differential equation (3.27) for the above H. we set Then x= (x1,x2, x3) is a branched surface such that the mean curvature is H and the normal Gauss map of x is G1. Moreover, if(?), then x is a regular surface.In the second part (Chapter 4 and Chapter 5), we obtain various versions of Omori-Yau's maximum principle on complete properly immersed submanifold with controlled mean curvature of complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay, and on complete properly immersed submanifold with controlled mean curvature of certain product manifolds. We also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature. Using the gener-alized maximum principle we give an estimate of the mean curvature of properly im-mersed submanifolds with bounded projection in Ni in the product manifold N1×N2:Theorem 3 ([38]) Let N1,N2 be complete Riemannian manifolds of dimen-sion n1, n2 respectively and the radial sectional curvature of N2 satisfy (?) (?), where c is a positive constant,ρ2 is the distance function from a fixed point on N2. Letφ:Mk→N1×N2 be an isometric immersion of a complete Riemannian manifold of dimension k> n2 with mean curvature vector H. Given q∈M, p=π1(φ(g))∈N1. Let BN1(r) be the geodesic ball of N1 centered at p with radius r. Assume that the radial sectional curvature (?) along the radial geodesics issuing from p is bounded as (?) in BN1(r). Suppose thatφ(M) (?) BN1(r)×N2 for (?), where we replace (?).(1)Ifφ:Mk→N1×N2 is proper, then(2) If then M is stochastically incomplete, where Cb is defined in the beginning of§4. We also give other applications of the generalized maximum principles.